How many 3-digit numbers that are divisible neither by 4 nor by 6 are there?

(A) 825 (B) 720 (C) 699 (D) 600 (E) None of the above

There are 900 3-digit numbers.

900/4 = ?
900/6 = ?
But if you count all those, you have counted twice all those that are multiples of 24, so subtract those off.
Now you have a count of all those divisible by 4 or 6 or both.
Subtract that from 900.

To find the number of 3-digit numbers that are divisible neither by 4 nor by 6, we can subtract the number of 3-digit numbers that are divisible by either 4 or 6 from the total number of 3-digit numbers.

Step 1: Find the total number of 3-digit numbers
To find the total number of 3-digit numbers, we need to find the range of 3-digit numbers. The range of 3-digit numbers is from 100 to 999 (inclusive).
So, the total number of 3-digit numbers is given by the formula:
Total = (999 - 100) + 1 = 900

Step 2: Find the number of 3-digit numbers divisible by 4
To find the number of 3-digit numbers divisible by 4, we need to determine the number of multiples of 4 within the range of 100 to 999.
Dividing 100 by 4, we get the first multiple of 4 (which is 25).
Dividing 999 by 4, we get the last multiple of 4 (which is 249).
So, the number of 3-digit numbers divisible by 4 is given by the formula:
Divisible by 4 = (249 - 25) + 1 = 225

Step 3: Find the number of 3-digit numbers divisible by 6
To find the number of 3-digit numbers divisible by 6, we need to determine the number of multiples of 6 within the range of 100 to 999.
Dividing 100 by 6, we get the first multiple of 6 (which is 16.666..., rounding up to 17).
Dividing 999 by 6, we get the last multiple of 6 (which is 166.5, rounding down to 166).
So, the number of 3-digit numbers divisible by 6 is given by the formula:
Divisible by 6 = (166 - 17) + 1 = 150

Step 4: Find the number of 3-digit numbers divisible by both 4 and 6 (which need to be subtracted)
To find the number of 3-digit numbers divisible by both 4 and 6, we need to determine the number of multiples of their least common multiple, which is 12.
Dividing 100 by 12, we get the first multiple of 12 (which is 8.333..., rounding up to 9).
Dividing 999 by 12, we get the last multiple of 12 (which is 83.25, rounding down to 83).
So, the number of 3-digit numbers divisible by both 4 and 6 is given by the formula:
Divisible by both 4 and 6 = (83 - 9) + 1 = 75

Step 5: Calculate the final answer
Now, we can find the number of 3-digit numbers that are divisible neither by 4 nor by 6:
Divisible neither by 4 nor by 6 = Total - (Divisible by 4 + Divisible by 6 - Divisible by both 4 and 6)
Divisible neither by 4 nor by 6 = 900 - (225 + 150 - 75)
Divisible neither by 4 nor by 6 = 900 - (375 - 75)
Divisible neither by 4 nor by 6 = 900 - 300
Divisible neither by 4 nor by 6 = 600

Therefore, there are 600 3-digit numbers that are divisible neither by 4 nor by 6.